For the rice: Mix together vinegar, sugar, and salt.
Rinse the rice in a fine-mesh sieve and drain. Put the rice and water in medium saucepan with a tight-fitting lid. Bring to a boil over high heat, reduce heat to very low and simmer, tightly covered. (For a good seal wrap the pan's lid tightly with a small kitchen towel and cover the saucepan. Make sure the towel's edges are folded up well away from the heat.) Cook until all the water is absorbed, about 15 minutes. Remove the rice from the heat (don't uncover) and set aside for 10 minutes.
Fluff the rice with a fork and transfer it to a large bowl. Add the vinegar mixture and toss with the rice, using a wooden spoon. Spread it out on a parchment paper-lined baking sheet. Fan the rice continuously with a fan or magazine until cooled. Cover the rice with a damp towel.
Meanwhile, beat the eggs with a teaspoon of water and a pinch of salt. Heat a nonstick skillet over medium heat, add the vegetable oil. Pour in the eggs, tilting the skillet as you add the eggs to make an even coating. Cook eggs until set. Invert eggs onto a cutting board and cut into strips.
To assemble rolls: Place the sushi mat with the slats running crosswise directly in front of you on a clean work space. Lay a sheet of nori shiny-side down and lined up with the closest edge of the mat. Lightly moisten fingers with water, and evenly press about 1/4 of the rice onto the nori, with 1 1/2-inches uncovered at the far edge. Line up the fillings about 1-inch from the closest near edge, evenly across the rice. Leave 1-inch of rice uncovered at the far edge.
Beginning with the near edge of the mat, tightly roll up the nori, rice, and fillings into a cylinder. Firmly tug on the rounded mat over the roll as you pull on the far edge of the mat to tighten the roll. Open the mat, dab unsealed edge of nori with a bit of water and roll the sushi forward to seal. Transfer the sushi to a plate and cover with damp paper towels. Repeat with remaining rice and fillings.
Cut each sushi roll, crosswise, into 8 pieces with a sharp wet knife.
Serve with bowl of soy sauce for dipping.
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